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On the time distribution of Earth’s magnetic
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field reversals
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Cosme F. Ponte-Neto1,*, Andrés R. R. Papa1,2
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Observatório Nacional, Rua General José Cristino 77, São Cristóvão,
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Rio de Janeiro, 20921-400 RJ, BRASIL
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Instituto de Física, Universidade do Estado do Rio de Janeiro, Rua São Francisco Xavier 524, Maracanã,
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Rio de Janeiro, 20550-900 RJ, BRASIL
Abstract
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This paper presents an analysis on the distribution of periods between
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consecutive reversals of the Earth’s magnetic field. The analysis includes the
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randomness of polarities, whether the data corresponding to different periods
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belong to a unique distribution and finally, the type of distribution that data obey. It
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was found that the distribution is a power law (which could be the fingerprint of a
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critical system as the cause of geomagnetic reversions). For the distribution
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function a slope value of –1.42 ± 0.19 was found. This value differs about 15%
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from results obtained when the present considerations are not taken into account
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and it is considered the main finding.
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Keywords
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geomagnetic, reversals, statistical test, distribution functions
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Introduction
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Geomagnetic reversals (periods during which the geomagnetic field swap
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hemispheres) are, together with the magnetic storms (because of the immediate
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effects on man’s activities), the most dramatic events in the magnetic field that we
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can measure at the Earth’s surface (Merrill, 2004). The time between consecutive
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geomagnetic reversals has typical values that range from a few tens of thousands
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of years to around forty millions of years while magnetic storms have durations of
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approximately two days. They also have different sources, while the magnetic
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storms are mostly associated to phenomena in the Sun and the terrestrial
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ionosphere, geomagnetic reversals are associated with changes in the Earth’s
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dynamo. Towards a deeper understanding of the laws that follow the geomagnetic
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reversals is devoted this work.
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Another common feature of both short period and long period of time
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phenomena is the appearance of power laws in their relevant distributions (Papa et
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al., 2006; Seki and Ito, 1993). One of the possible mechanisms that produce power
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law distributions for, for example, the distribution of times between consecutive
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periods of great activity, is the mechanism of self-organized criticality or, more
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specifically, of threshold systems. It is quite remarkable that, phenomena
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essentially diverse (like magnetic storms and geomagnetic reversals), could be
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sustained by similar types of mechanisms.
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Threshold systems are the base for the behavior of many dissimilar
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phenomena. They are composed by elements that behave in a special manner: 1)
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the elements are able to store potential energy up to a given threshold; 2) they are
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continuously supplied with potential energy; 3) when the accumulate potential
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energy in an element reach the threshold part of its energy is released to neighbor
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elements and out of the system; 4) eventually, the energy released to some of the
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neighbors will be enough to surpass its own threshold; 5) this element will release
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part of its energy to the neighborhood and out of the system and so on. In this way
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a single element can spark a long chain reaction that will extinguish only when all
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the elements are below the threshold. At a first reading the concept of threshold
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system could appear very abstract, but there are some simple examples that can
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help in demystifying the concept. Suppose that we locate a block of wood on a
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surface and attach a spring to it. If we try to move the block by pulling the opposite
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extreme of the spring initially it will not move. The block will move only when the
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potential energy accumulated in the spring reach the static friction. In this case the
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static friction plays the role of threshold. The released energy (as we are
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considering a single block-spring set there are no neighbors) is composed by the
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thermal energy (produced by the dynamic friction between the block and the
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surface) and acoustic energy (the noise that the block produces while sliding on
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the surface). Actually, models with systems of many spring and interconnected
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block have served to reproduce some of the main characteristics of earthquakes.
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The energy has to be supplied at a low rate (compared to the maximum power that
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the system can dissipate) otherwise there would be no avalanches. In the block-
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spring example, if we pull the spring very rapidly (i.e. if we introduce energy at a
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high rate) probably the block will never stop once in movement. It is a usual (non
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exclusive) signature of self-organized criticality and threshold systems the
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appearance of power laws
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f(x) = c . xd
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where x is the variable, c is some proportionality constant, f(x) is the distribution of
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the variable x and d is the exponent. These concepts will help us in the
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interpretation of some of the results that we will describe.
(1)
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Works devoted to the study of the time distribution of geomagnetic reversals
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include, among many others, an analysis of scaling in the polarity reversal record
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(Gaffin, 1989), a search for chaos in record (Cortini and Barton, 1994), a critical
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model for this problem (Seki and Ito, 1993) and more recently, a long-range
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dependence study in the Cenozoic reversal record (Jonkers, 2003). Gaffin (1989)
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pointed out that long-term trends and non-stationary characteristics of record could
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difficult a formal detection of chaos in geomagnetic reversal record. It is our opinion
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that because of this and also because the low number of reversals, in the work of
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Gaffin actually, it was pointed out that it would be very difficult to detect in a
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consistent manner that the geomagnetic reversals present any characteristic at all,
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without mattering which this characteristic could be (including chaos).
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Our study differs from those works in that, we explore the equivalence of
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both polarizations through some well-known non-parametric test on the reversals
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time series. We then study the possibility of diverse periods pertain to the same
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distribution and finally the distribution that geomagnetic reversal effectively follows.
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Our work is closer to the one by Jonkers (2003) and in some sense complements
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it.
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Analysis
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There is some recent evidence (Clement, 2004) on a dependence of the
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geomagnetic polarity reversals on the site where the analyzed sediments are
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collected. This can be the fingerprint of higher order (not only dipolar) contributions
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to the components of the Earth’s magnetic field. We have not considered those
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variations. Another feature that was not considered by us are the detailed
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variations of the Earth’s dipole (Valet et al., 2005). We have just considered
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polarity inversions. We used the more complete data that we have found (Cande
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and Kent, 1992, 1995).
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In Figure 1 we present the sequences of reversals during the last 120 My. It
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can be seen a clear difference between the periods 0-40 My and 40-80 My, before
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the great Cretaceous isochrone. Our intuitive reasoning can be further supported
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by some evidences of tectonic changes experienced by the Earth at the same
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epoch (around 40 My ago) that could have influenced the dynamo system: the
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change in direction of growth of the Hawaiian archipelago. Those are the reasons
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to study separately, at least initially, both periods.
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We wonder now, are both polarizations in each of the periods equally
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probably? If both polarizations are equivalent this is a useful fact from the statistical
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point of view. Instead of two small samples we have a single and larger one. At the
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same time, the equivalence might be pointing to an almost inexistence of tectonic
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influence on the reversal rate because the Earth has a defined rotation direction
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(although the rotation is considered a necessary condition). On the other hand, an
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almost inexistent influence (or very small influence) is compatible with the
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requirement of self-organized criticality and threshold systems of a small energy
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deliver rate. However, see below.
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We have implemented a non-parametric sequence u test. To do so we have
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taken the shortest interval in each period between consecutive geomagnetic
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reversals as a trial (0.01 My and 0.044 My for 0-40 and 40-80 My periods,
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respectively). We normalized to this value the rest of the reversals in each period.
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The result (rounded) was taken as a sequence of identical consecutive trials for
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that polarization. In this way we obtained a sequence of the type (N means normal
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and R means reverse polarization) “NNNRRNNNNNRRRNRNNN …”, over which
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we implemented the test. For the period 0-40 My, that includes around 140
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reversals, it was obtained that both polarizations are almost identically probable
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(1966 trials in one polarization against 1985 in the opposite one). On the other
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hand, for the period 40-80 My, that contains only 40 reversal, the result was no so
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good: for one polarization we obtained 632 trials while for the opposite one only
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353 trials. There are two possible explanations for this fact: there was some factor
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that favored a polarization over the other (of tectonic nature, for example) or the
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sample is not large enough to avoid fluctuations (note that the number of reversals
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in the 40-80My period is around 25% the number in the 0-40 My period). We will
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assume that the second explanation is the actual one. There are no reasons to
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believe that the mechanism producing the reversals has changed in nature.
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Consequently, for each of the periods both reversal polarities have been
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considered as a single sample. The other relevant result that we can extract from
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the trials is that we must reject the null hypotheses H0 of randomness almost with a
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100% confidence. This result coincides with a previous one (Jonkers, 2003), but to
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arrive to that conclusion there were used specific methods (aggregate variance
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and absolute value) devised for long-range-dependences studies.
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A natural question that arises is, do both periods correspond to the same
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distribution? Before trying to answer this question let us make some considerations
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on distribution functions. From a “classical” point of view, belong-to-the-same-
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distribution means to have similar means and standard deviations (this assertion
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includes many distribution function types like gaussians, lorentzians, etc.). When
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we work with power-law distribution functions special cares have to be taken
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because the distribution are endless. This can be easily seen in a log-log plot. In
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this type of plot the distribution takes the form of a straight line. So, belong-to-the-
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same-distribution could well mean that both data sets fit the same straight line but
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in different intervals. To try answering the question we separately present in Figure
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2 the frequency distribution of reversals for the two periods using log-log scale and
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logarithmic bins. Both distributions present approximately a top-of-a-bell shape but
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with maximum at different values of time. Logarithmic bins constructions have the
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property of converting exact (functional) power-law distribution functions with
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exponent d, in power laws with exponent d+1. At the same time, if there is a
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reasonable number of data, they produce best quality (soft) curves because they
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average (integrate) over increasing windows. From Figure 2 it can be seen that for
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small time periods both curves initially grow (which means that the distribution, if
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following some power law, presents an exponent d ≥ 0). For the highest values
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(again, if following a power law) the exponent is d < -1 (because for d = -1 the
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logarithmic bin plots would be constant values). However, the number of points is
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not large enough for more accurate predictions on the exponents from this type of
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graph.
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previous works we believe to be a power law with a unique slope). Supposing that
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they effectively follow a power law then they also should rest approximately on a
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single straight line: fortunately, we should not be worried with the weight of each of
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the periods because the time (which means, statistical weight) is approximately the
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same for both periods. However, this poses a problem to construct a single
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histogram with both periods (i.e., to consider both periods as part of a single
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sample): the middle values could be counted twice while the extremes just once. In
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order to compare considering both periods as a single sample or as two separate
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samples we constructed the frequency distribution from the whole period from 0 to
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80 My. Figure 3 shows the result. A linear fit to the data gave a value of –1.64 ±
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0.24 for the slope. We have then constructed independently the frequency
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distribution for each of the periods and represented them in a single plot. The result
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is shown in Figure 4. The slope of the linear fit to both data takes a value –1.42 ±
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0.19, well apart from the result that we have found when not taking into account
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our present considerations (however, within the error interval). The most accepted
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value for this slope is ~ –1.5, near the average of the two that we have found.
From the shape we deduce that they follow the same law (following
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Self-organized systems have no a typical time scale nor a typical length
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scale (and the behavior in time and in space are closely related, both are fractals).
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The unique relevant length is the system size. The same model system with
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different sizes gives results that depend on the size in the way we explain now. As
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an example let us take a simple model for the brain (Papa and da Silva, 1997). If
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we simulate the model using 1024 elements we will obtain power law distributions
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for the first return time with a slope of –1.58. If we now use 4096 elements we will
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obtain the same power law dependence with the same slope. The difference
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between both cases is that while for the case of 1024 elements we obtain “clean”
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power laws for about two decades, when we use 4096 we can extend this interval
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to around four decades. So, different intervals in the same power law could
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indicate different sizes of activity regions for geomagnetic currents. Besides the
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fact of having small samples, this is a factor that could partially explain why the
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distributions go in the form of a power law to lower or higher values, to the right.
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Another factor that can limit the extension of power laws by the left (small values)
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is the rate at which energy is delivered to the system. It is a threshold (can not be
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confused with the threshold mentioned at the introductory section) for the smaller
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avalanches that exist and can be observed. In this way it should increase the
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average value between consecutive avalanches or, in other words, will cause an
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increase in first return times (the equivalent of reversals for the present problem).
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Conclusions
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Using classical statistical analysis we have excluded the possibility of
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reversals be a random process (or the result of a random process), conclusion that
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coincides with previous ones demonstrated through different methods. From the
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period 0-40 My (and in a less degree, from the period 40-80 My), where the
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probability of both polarities was almost identical, we can conclude that the
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influence of the geodynamo on reversals is null or very small. This fact is
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compatible with the necessity for self-organized criticality and threshold systems of
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a small energy release rate. From our results we can also conclude that the
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existence of power laws in the time distribution of geomagnetic reversals is a
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probable fact. The existence of power laws can be the result of many mechanisms.
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So, our results do not demonstrate the existence of a critically self-organized (or
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even a simple critical) system as the source for geomagnetic reversals but they are
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compatible with these possibilities. The value of –1.42 for the slope of the
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distribution function is an original finding and needs further confirmation by other
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authors. Modeling of the source system for reversals is an exciting problem. Some
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works are currently running with this aim and will be published elsewhere.
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Acknowledgements
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The authors sincerely acknowledge partial financial support from FAPERJ
(Rio de Janeiro Founding Agency) and CNPq (Brazilian Founding Agency).
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References
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Cande, S. C., Kent, D. V., 1992, A new geomagnetic polarity time scale for the late
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Cretaceous and Cenozoic, Journal of Geophysical Research 97, No. B10, 13917-
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13951.
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Cande, S. C., Kent, D. V., 1995, Revised calibration of the geomagnetic polarity
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time scale for the late Cretaceous and Cenozoic, Journal of Geophysical Research
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100, No. B4, 6093-6095.
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Clement, B. M., 2004, Dependence of the duration of geomagnetic polarity
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reversals on site latitude, Nature 428, 637-640.
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Cortini, M., Barton, C., 1994, Chaos in geomagnetic reversal records: A
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comparison between Earth’s magnetic field data and model disk dynamo data,
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Journal of Geophysical Research 99, No. B9, 18021-18033.
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Gaffin, S., 1989, Analysis of scaling in the geomagnetic polarity reversal record,
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Physics of the Earth and Planetary Interiors 57, 284-290.
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Jonkers, A. R. T., 2003, Long-range dependence in the Cenozoic reversal record,
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Physics of the Earth and Planetary Interiors 135, 253-266.
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Merril, R. T., 2004, Time of reversal, Nature 428, 608-609.
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Papa, A. R. R.; Barreto, L. M.; Seixas, N. A. B., 2006, Statistical Study of Magnetic
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Disturbances at the Earth’s Surface,
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Terrestrial Physics (to appear).
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Papa, A. R. R., da Silva, L., 1997, Earthquakes in the brain, Theory in Biosciences
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116, 321-327.
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Seki, M., Ito, K., 1993, A phase-transition model for geomagnetic polarity reversals,
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Journal of Geomagnetism and Geoelectricity 45, 79-88.
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Valet, J.-P., Meynadier, L., Guyodo, Y., 2005, Geomagnetic dipole strength and
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reversal rate over the past two million years, Nature 435, 802-805.
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Figure Captions
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Figure 1.- Representation of geomagnetic reversals from 120 My ago to our days.
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We arbitrarily have assumed –1 as the current polarization.
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Figure 2.- Log-log plot of the distributions of intervals between consecutive
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reversals for the periods from 0 to 40 My (squares) and from 40 to 80 My (circles).
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We have used logarithmic bins of size 0.015x2n My, where n=0, 1, 2, 3, 4, 5, 6 and
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7. To highlight the similarity between both curves they were normalized to have
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approximately the same height.
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Figure 3.- Frequency distribution for the period from 0 to 80 My. The bold straight
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line is a linear fit to the data. It has a slope –1.64 ± 0.24.
Journal
of
Atmospheric and Solar
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Figure 4.- Frequency distributions for the periods from 0 to 40 and from 40 to 80
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My. From left to right the points of numbers 1, 3, 4, and 6 belong to the period from
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0 to 40 My. Points with number 2, 5, 7, 8 and 9 belong to the 40 – 80 My period.
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The bold straight line is a simultaneous linear fit to both data. It has a slope value
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of –1.42 ± 0.19.
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Figure 1
Polarization (a.u.)
1,0
0,5
0,0
-0,5
-1,0
0
20
40
60
80
100
120
Age(My)
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Figure 2
Frequency
10
1
0 ,1
1
10
T im e b e tw e e n re v e rs a ls (0 .0 1 M y)
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100
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Figure 3
Frequency
100
10
1
0 ,1
1
10
A g e (M y )
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Figure 4
Frequency
100
10
1
0 ,1
1
A g e (M y )
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